03hw - Solutions to problem set 3 (141A Sp07) 1 1. (ISSP...

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Unformatted text preview: Solutions to problem set 3 (141A Sp07) 1 1. (ISSP 3.3) Basis of two unalike atoms. At k = / a , equation (20) becomes 2- 2 M 1 C ! u = 2- 2 M 2 C ! v = Note that the u and v sub-lattices are decoupled at this value of k ; u can change without changing v , and vice versa. There are two solutions to the above equations: u = 0, 2 = 2 C / M 2 and v arbitrary; and v = 0, 2 = 2 C / M 1 and u arbitrary. 2. (ISSP 3.4) Kohn anomaly. Equation (16a) gives 2 = 2 A M X p = 1 sin pk a pa (1- cos( pka )) = 2 A Ma X p = 1 sin pk a p- 1 2 sin p ( k + k ) a p- 1 2 sin p ( k- k ) a p ! Gradshteyn and Ryzhik ( Table of integrals, series and products )1.441 gives X p = 1 sin p p = - 2 < < 2 (you can prove this by writing the sin as the sum of two exponentials, and then using the taylor expansion of log (1 + x ).) So for k > k , we have 2 = 2 A Ma X p = 1 sin pk a p- 1 2 sin p ( k + k ) a p- 1 2 sin p ( k- k ) a p ! = 2 A Ma - k a 2- 1 2 - ( k + k ) a 2- 1 2 - ( k- k ) a 2 !...
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03hw - Solutions to problem set 3 (141A Sp07) 1 1. (ISSP...

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